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A096822 Smallest primes of form p=2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of primes. 2
3, 5, 3, 549755813881, 7, 5, 3, 17, 47, 13, 11, 41, 7, 5, 3, 97, 31, 29, 2011, 89, 23, 536870869, 19, 17, 79, 13, 11, 73, 7, 5, 3, 193, 191, 61, 59, 953, 439, 53, 179, 433, 47, 173, 43, 41, 167, 37, 163, 929, 31, 29, 67108763, 409, 23, 149, 19, 17, 911, 13, 11, 137 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If 2n-1 is a provable Riesel number (A101036), then there exists a finite set of primes P(2n-1) such that every 2^x-(2n-1) > 0 is divisible by p(x) in P(2n-1). If some 2^x-(2n-1) = p(x), then a(n) = p(x). Otherwise, p(x) is a proper divisor of 2^x-(2n-1), which must be composite, and no a(n) exists.

For example, if n = 254602, then 2n-1 = 509203 is a provable Riesel number. Every 2^x-509203 > 0 is divisible by prime p(x) in P(509203) = {3,5,7,13,17,241}.  2^x-509203 > 0 implies x >= 19 implies 2^x-509203 > 241 >= p(x), so p(x) is a proper divisor and every 2^x-509203 is composite. Hence a(254602) does not exist.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..935

EXAMPLE

a(1) = 3 is the first Mersenne-prime;

a(64) = 2^47-127 = 140737488355201, where 47 = A096502(64), 127=2*64-1.

MATHEMATICA

f[n_]:=Module[{lst={}, exp=Ceiling[Log[2, 1+n]]}, While[!PrimeQ[2^exp-n], exp++]; AppendTo[lst, 2^exp-n]]; Flatten[f/@Range[1, 1001, 2]] (* Ivan N. Ianakiev, Mar 08 2016 *)

CROSSREFS

Cf. A096502.

Sequence in context: A249908 A195418 A065974 * A195383 A086567 A016611

Adjacent sequences:  A096819 A096820 A096821 * A096823 A096824 A096825

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 13 2004

STATUS

approved

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Last modified August 17 11:20 EDT 2017. Contains 290635 sequences.